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โš™๏ธย ap physics c: mech

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๐ŸŽณย Unit 4

ย ย โ€ขย ย โฑ๏ธ2 min read

4.1 Center of Mass ๐Ÿ’

Daniella Garcia-Loos

daniella garcia-loos

โฑ๏ธ November 5, 2020

๐Ÿ“…

4.1: Center of Mass ๐Ÿ’

The center of mass, also sometimes called the center of gravity, is typically what we refer to as the geometric position in an object defined by: the mean position of every section of the object or system, weighted by mass. In other words, this is a place where the object is balanced in our gravitational field.

Below you can see an example of finding the center of mass in the x direction of a system of masses:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-y8IN4zfze7lh.png?alt=media&token=c7709ed2-6ae3-438c-a180-fc68ac3317d1

Image from LibreTexts

For a system of masses:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-bC6XFz44KSSp.png?alt=media&token=dda3a255-4cde-4d88-a9a4-70de518f8182

Calculus definition:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-OOocWGoTIcbc.png?alt=media&token=5eb765b2-9ba6-4cef-ab7c-2c261c67d464

Another way to format the above formula is with linear mass density:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-HIod6BMlq2mc.png?alt=media&token=f3b9649a-9f36-4b94-b57c-efda789b05d5

Linear mass density is typically a constant for something that is uniform, so it can be found with an equation like:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-eZwxTNmKMseX.png?alt=media&token=55c28492-fc1f-450a-88e7-f63ba2714d91

However, since AP loves to make us do calculus, we will sometimes see non-uniform objects! This means that the linear mass density would be a function.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-jXSWJWxNYCf2.png?alt=media&token=6517fe33-8b56-43c5-8119-3b5aa828f52c

Let's try to calculate the center of mass of a uniform rod!

We can begin with one of the formulas we discussed above and place bounds on it:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-x03525VJLCvI.png?alt=media&token=9134e4ca-0eab-4dbb-9b24-2a7487c149d4

Since we know that the rod is uniform, we can take the linear mass density out of the integral because it is a constant.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-T8SR7d2YAyGL.png?alt=media&token=4eb29bf4-7ea7-405e-ad8b-999fca12bc7e

As you can see, the lambdas cancel out! Now we can evaluate the integrals.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-p1KaZ3IqoouN.png?alt=media&token=07ccb966-5cda-45c4-bc3b-5c0b9388ccc1

Now we can plug in our bounds and simplify. This leads us to:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-tRnYCZH5EMdK.png?alt=media&token=c27237b0-aa61-44ec-bb7f-b60d4ad853a6

Hopefully, this answer seems intuitive to you! We'll be seeing problems similar to this when we tackle rotational inertia next unit.

Practice Questions

Even though AP Physics 1 is not calculus-based, we can practice applications of the center of mass with FRQs from that test too!

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-eM7ypWB4V62L.png?alt=media&token=d35a25ce-be14-4f73-b247-6e74bb27e309

Taken from College Board

Answer:

The trick to this question is realizing that it is asking for the center of mass of the system. So the speed of it should only change when momentum isn't conserved, meaning when there is impulse!

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-VT3ALUaA5jeQ.png?alt=media&token=277b626f-d677-4b4e-a716-882cf467d184


https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-hKW9AKibyOLB.png?alt=media&token=fe7084ec-0e41-4a66-aee4-3a33dbdd5391

Answer:

Same focus as before, realize it is the center of mass of the system! Think of how you were searching for the x coordinate of the center of mass, you can apply the same strategy for velocity.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-0T9xBLuc6yYC.png?alt=media&token=72862d41-f87a-44ea-95db-8b6a30e2a178

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